| L(s) = 1 | + 1.12·2-s + 3-s − 0.743·4-s + 5-s + 1.12·6-s + 2.49·7-s − 3.07·8-s + 9-s + 1.12·10-s + 5.60·11-s − 0.743·12-s + 3.63·13-s + 2.79·14-s + 15-s − 1.96·16-s + 1.30·17-s + 1.12·18-s + 7.43·19-s − 0.743·20-s + 2.49·21-s + 6.28·22-s + 0.358·23-s − 3.07·24-s + 25-s + 4.07·26-s + 27-s − 1.85·28-s + ⋯ |
| L(s) = 1 | + 0.792·2-s + 0.577·3-s − 0.371·4-s + 0.447·5-s + 0.457·6-s + 0.942·7-s − 1.08·8-s + 0.333·9-s + 0.354·10-s + 1.68·11-s − 0.214·12-s + 1.00·13-s + 0.746·14-s + 0.258·15-s − 0.490·16-s + 0.317·17-s + 0.264·18-s + 1.70·19-s − 0.166·20-s + 0.543·21-s + 1.33·22-s + 0.0747·23-s − 0.627·24-s + 0.200·25-s + 0.798·26-s + 0.192·27-s − 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.827873565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.827873565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 5.60T + 11T^{2} \) |
| 13 | \( 1 - 3.63T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 7.43T + 19T^{2} \) |
| 23 | \( 1 - 0.358T + 23T^{2} \) |
| 29 | \( 1 - 3.27T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 - 2.16T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 0.571T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 5.93T + 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 4.18T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.85T + 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 - 4.94T + 89T^{2} \) |
| 97 | \( 1 + 0.958T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.139628233377434871033929351268, −7.36402936536415115062425580934, −6.49745691034320278281403078368, −5.82426344944060874126751516904, −5.09953512915855868931349309559, −4.43709351960731522849444637997, −3.54075487133303023057567470768, −3.24539474368590097691230136036, −1.76303495775586773550169739911, −1.15318287634826843889295828979,
1.15318287634826843889295828979, 1.76303495775586773550169739911, 3.24539474368590097691230136036, 3.54075487133303023057567470768, 4.43709351960731522849444637997, 5.09953512915855868931349309559, 5.82426344944060874126751516904, 6.49745691034320278281403078368, 7.36402936536415115062425580934, 8.139628233377434871033929351268
